Theorems stating the existence of an object—such as the solution to a problem or equation—are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations. The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, they obtain a logically coherent discussion for students at a specific phase of their mathematical development. The treatment begins with a survey of fundamental existence theorems and advances to general existence and uniqueness theorems. Subsequent chapters explore the Picard iterants, properties of solutions, and linear differential equations.